Let $M=(M_t)_{t \geq 0}$ be a Martingale with respect to the filtration $\mathcal{F}=(\mathcal{F}_t)_{t \geq 0}$. Assume that $\mathbb{E}(M_t^2)<\infty$ for all $t \geq 0$. Let $0=t_0<t_1<...<t_N=T$ be a partition of interval $[0,T]$. Further, let $(\xi_n)_{n=0,1,...,N}$ be a family of random variables such that $\xi_0 \in \mathcal{F}_{t_{0}}$ and $\xi_n \in \mathcal{F}_{t_{n-1}}$ for all $n \geq 1$, and such that $\mathbb{E}(\xi_n^2)<\infty$ for all $n \geq 0$.
Consider the process $X=(X_t)_{t \in [0,T]}$ given by:
$$ X_t=\xi_0+\sum_{n=1}^{\{t\}}\xi_n(M_{t_n}-M_{t_{n-1}}) $$
Where $\{t\}$ is defined as $k-1$ if $t \in (t_{k-1},t_k]$
The problem is to prove the above process is a Martingale with respect to $\mathcal{F}$. I have already proven the first condition (measurability), but I am having trouble proving integrability and the Martingale property. Any help would be much appreciated.
For the integrability use the triangle and Hölder inequality. It should be straight forward.
For the Martingale property use that for all $t_{n-1}<t$ $\xi_{n}$ is $\mathcal{F}_{t}$ measureable, so by using the tower property of conditional expectations a finite number of times, you can pull it out of the conditional expectation and by the matringale property of $M$ the most inner difference equals zero. That's why all terms with higher index disappear.
Sry, for the short answer so far. If it is not sufficent, I can answer in more detail later.